Quantum walks are a popular area of study in quantum computing, as they have shown success in aiding the construction of efficient algorithms [1]}.
Reference [2]} gives an excellent introduction to the concept of quantum walks.
In essence, they are stochastic processes that, when evolved and measured, give information about the system in which they have evolved.
The structure of the evolution and the properties measured can be adjusted to change the information obtained.
We are interested in the walk procedure developed in [3]}, which considers the problem of element-distinctness.
This problem is quite far removed from our current study, but it is important for the fact that it introduces a particular walk operator based on Grover diffusion operators.
Following this research, [4]} further investigated diffusion-based quantum walks defined by probabilistic maps, and considered the spectra of the walk operators.
Finally, [5]} expanded on these findings from the perspective of Hamiltonian simulation, and showed that the walk operators could be defined from a Hamiltonian to produce an operator with a spectrum closely
related to that of the Hamiltonian itself.
Reference [6]} further explored methods by which these walk operators could be implemented in practice.
